Improved Numerical And Numeric-Analytic Schemes In Nonlinear Dynamics And Systems With Finite Rotations

Abstract

This thesis deals with different computational techniques related to some classes of nonlinear response regimes of engineering interest. The work is mainly divided into two parts. In the first part different numeric-analytic integration techniques for nonlinear oscillators are developed. In the second part, procedures for handling arbitrarily large rotations are addressed and a few novel developments are reported in the process. To begin the first part, we have proposed an explicit numeric-analytic technique, based on the Adomian decomposition method, for integrating strongly nonlinear oscillators. Numerical experiments suggest that this method, like most other numerical techniques, is versatile and can accurately solve strongly nonlinear and chaotic systems with relatively larger step-sizes. It is then demonstrated that the procedure may also be effectively employed for solving two-point boundary value problems with the help of a shooting algorithm. This has been followed up with the derivation and numerical exploration of variants of a recently developed numeric-analytic technique, the multi-step transversal linearization (MTrL), in the context of nonlinear oscillators of relevance in engineering dynamics. A considerable generalization and improvement over the original form of a MTrL strategy is achieved in this study. Finally, we have used the concept of MTrL method on the nonlinear variational (rate) equation corresponding to a nonlinear oscillator and thus derive another family of numeric-analytic techniques, presently referred to as the multi-step tangential linearization (MTnL). A comparison of relative errors through the MTrL and MTnL techniques consistently indicate a superior quality of approximation via the MTrL route. In the second part of the thesis, a scheme for numerical integration of rigid body rotation is proposed using only rudimentary tensor analysis. The equations of motion are rewritten in terms of rotation vectors lying in same tangent spaces, thereby facilitating vector space operations consistent with the underlying geometric structure of rotation. One of the most important findings of this part of the dissertation is that the existing constant-preserving algorithms are not necessarily accurate enough and may not be ideally applicable to cases wherein numerical accuracy is of primary importance. In contrast, the proposed rotation-algorithms, the higher order ones in particular, are significantly more accurate for conservative rotational systems for reasonably long time. Similar accuracy is expected for dissipative rotational systems as well. The operators relating rotation variables corresponding to different tangent spaces are also investigated and this should provide further insight into the understanding of rotation vector parametrization. A rotation update is next proposed in terms of rotation vectors. This update, employed along with interpolation of relative rotations, gives a strain-objective and path independent finite element implementation of a geometrically exact beam. The method has the computational advantage of requiring considerably less nodal variables due to the use of rotation vector parametrization. We have proposed a new isoparametric interpolation of nodal quaternions for computing the rotation field within an element. This should be a computationally efficient alternative to the interpolation of local rotations. It has been proved that the proposed interpolation of rotation leads to the objectivity of strain measures. Several numerical experiments are conducted to demonstrate the frame invariance, path-independence and other superior aspects of the present approach vis-`a-vis the existing methods based on the rotation vector parametrization. It is emphasized that, in order to develop an objective finite element formulation, the use of relative rotation is not mandatory and an interpolation of total rotation variables conforming with the rotation manifold should suffice.